symmetric monoidal (∞,1)-category of spectra
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
A homotopy $T$-algebra over a Lawvere theory $T$ is a model for an $\infty$-algebra over $T$, when the latter is regarded as an (∞,1)-algebraic theory.
As a model, homotopy $T$-algebras are equivalent to strict simplicial algebras.
For $T$ (the syntactic category of) a Lawvere theory with generating object $x$ an ordinary algebra over a Lawvere theory functor $T \to Set$ that preserves products, in that for all $n \in \mathbb{N}$ the canonical morphism
is an isomorphism.
A homotopy $T$-algebra is a functor $A : T \to$ sSet with values in Kan complexes such that for all $n \in \mathbb{N}$ this canonical morphism is a weak homotopy equivalence.
For $n \in \mathbb{N}$ write $F_T(n)$ for the free simplicial $T$-algebra on $n$-generators, which is the image of $x^n$ under the Yoneda embedding $j : T^{op} \to [T,sSet]$. (See Lawvere theory for more on this.)
A homotopy $T$-algebra is precisely
a fibrant object in the projective model structure on simplicial presheaves;
which is a local object with respect to the canonical morphisms
for all $n \in \mathbb{N}$.
The fibrant objects in $[T,sSet]_{proj}$ are precisely the Kan complex-valued co-presheaves. Because $F_T(n)$ is representable, it is cofibrant in $[T,sSet]_{proj}$ (as one easily checks). Therefore the derived hom-spaces between $F_T(\cdots)$ and a degreewise Kan complex-valued $A$ may be computed simply as the sSet-hom-objects of the simplicial model category $[T,sSet]$ and so the degreewise fibrant $A$ being a local object means that all morphisms of sSet-hom-objects
Due to the respect of the hom-functor for limits the expression on the right is
Using the Yoneda lemma the morphism in question is indeed isomorphic to
This observation motivated the following definition.
The model category structure for homotopy $T$-algebras is the left Bousfield localization $[T,sSet]_{proj,loc}$ of the projective model structure on simplicial presheaves $[T,sSet]_{proj}$ at the set of morphisms $\{\coprod_n F_T(1) \to F_T(b)\}_{n \in \mathbb{N}}$.
The model structure for homotopy $T$-algebra $[T,sSet]_{proj,loc}$ is a left proper simplicial model category.
Because the model structure on simplicial presheaves is and left Bousfield localization of model categories preserves these properties.
The inclusion
has a left adjoint
The limits in $T Alg$ are easily seen to be limits in the underlying sets. Hence $i$ preserves all limits. The statement then follows by observing that the assumptions of the special adjoint functor theorem are met:
$T Alg$ is complete;
it is a well powered category since $[T,Set]$ is and the subobject in $T Alg$ are special subobjects in $[T,Set]$;
it has a small cogenerating set given by the representables.
An explicit description of $F$ is around HTT, lemma 5.5.9.5.
Let $T Alg^{\Delta^{op}}_{proj}$ be the category of simplcial T-algebras equipped with the standard model structure on simplicial algebras (with weak equivalences and fibrations the degreewise weak equivalences and fibrations in simplicial sets).
The adjunction from the previous lemma
is a Quillen adjunction which is a Quillen equivalence
This is theorem 1.3 in (Badzioch)
The model structure on homotopy $T$-algebras for $T =$ CartSp the Lawvere theory of smooth algebras is considered in (Spivak) in the study of derived smooth manifold. (There is also a bit of disucssion of the relation to the model structure on simplicial algebras there.)
algebra over an algebraic theory
∞-algebra over an (∞,1)-algebraic theory
In
the model structure on homotopy $T$-algebras is discussed and its Quillen equivalence to simplcial $T$-algebras is proven.
A related discussion showing that simplicial $T$ algebras model all $\infty$-$T$-algebras is in
The model structure on homotopy $T$-algebras for $T =$ CartSp the Lawvere theory of smooth algebras is considered in